We obtain various upper bounds for the numerical radius w(T) of a bounded linear operator T defined on a complex Hilbert space H, by developing the upper bounds for the α-norm of T, which is defined as ‖T‖α=supα|⟨Tx,x⟩|2+(1-α)‖Tx‖2:x∈H,‖x‖=1 for 0≤α≤1. Further, we prove that (Formula presented.) For 0≤α≤1≤β, the operator T is called (α,β)-normal if α2T∗T≤TT∗≤β2T∗T holds. Note that every invertible operator is an (α,β)-normal operator for suitable values of α and β. Among other lower bounds for the numerical radius of an (α,β)-normal operator T, we show that (Formula presented.) where ℜ(T) and ℑ(T) are the real part and imaginary part of T, respectively.